Mathematics, Discovered or Invented?

[muhammad_isa]

Nope, the order of the deduction is the other way around. I first started thinking about how to verify beliefs as valid.Then I applied that to religious beliefs..

Hmm .. so how do you deduce that there is no such thing as "mathematical reality" outside of "brains" ?

You insist that it is reasonable to believe in something for which there is no evidence (as you admit)..

No PHYSICAL evidence, as such, no.. but NOT "no evidence" .. not "plucked out of thin air".

I have studied math for the last several decades..

.and I have been contemplating/studying religious belief over the last several decades..

I have been interested in the philosophical aspects of it for nearly that long. I have read Plato and Aristotle, as well as more modern philosophers (Putnam, for example). I have *concluded* that math is a cultural phenomenon, like language.

Mmm .. philosophy .. one can believe almost anything, as one can with religion.

 

[Polymath257]

Hmm .. so how do you deduce that there is no such thing as "mathematical reality" outside of "brains" ?

Because mathematics is a language. languages only exist inside of brains.

No PHYSICAL evidence, as such, no.. but NOT "no evidence" .. not "plucked out of thin air".

OK, what evidence do you have? From what I can see, it is originally 'plucked out of thin air' and then transmitted by culture.

.and I have been contemplating/studying religious belief over the last several decades..


Mmm .. philosophy .. one can believe almost anything, as one can with religion.

 

[muhammad_isa]

OK, what evidence do you have? From what I can see, it is originally 'plucked out of thin air' and then transmitted by culture.

Well, saying that just implies that revelations/scrolls that are claimed to have been from the One of God of Abraham, are all part of one gigantic conspiracy .. no?

You can answer in the context of the major revelations as a starting point.
i.e. the OT, NT & Qur'an

If not, I assume you have an agenda to confuse the issues raised, by bringing up every Tom, Dick & Harry.

 

[Heyo]

A bit, yes. Although it was, to some extent, a natural progression.

The question of whether math is invented or discovered is linked to the nature of abstractions: do they exist outside of our minds or not? if so, then they provide an example of non-physical existence. But, even if they only exist inside of our minds, they would give such an example if our minds are, themselves, non-physical. Also, even if abstractions do exist outside of our minds, there is the issue of how are minds are able to 'perceive' them. For some, this also suggests the possibility of a non-physical mind.

Nope, not for me. A mind is always a function of a brain. Just like a program is always a function of a computer. The program is an abstraction, even more so is the algorithm behind the program. It can be implemented in many languages and run on many different computers, but there always has to be some "brain". That doesn't make the algorithm none-existent or irrelevant. Abstractions exist, they just aren't real.

 

[RestlessSoul]

Another aspect that many people don't grasp is the concept of 'composed of' isn't as definite for composite quantum particles. So, for example, protons are usually said to be 'composed of' two up quarks and a down quark. But that isn't completely accurate.

The reason is that both the up and down quarks can 'change flavors' spontaneously through the weak force. This means that all protons have a 'presence' of strange quarks in their 'composition'. The same was recently found for charm quarks.

At the quantum level, composition, like most other 'properties' is a probabilistic thing. The classical idea of composition is simply not appropriate for quantum particles.

Finally, I am not sure why that would throw the 'concept of physical as fundamental' into question. Can you elaborate? In particular, how do you define the concept of 'physical' and 'fundamental'?

By physical, for the sake of this argument let’s say possessing definite material properties. And by fundamental, let’s say at the limit of reductionism; or, looked at from a different angle, that from which all things emerge.

If subatomic particles have properties only in the manner of their relation to other particles and to the forces governing those relations; and if they don’t even have continuous temporal trajectories, but only the probability of materialising at any given set of co-ordinates from one moment to another, can they be said to exist as entities at all? Or are they rather, spontaneous events (as Schrodinger described them) unfolding in web of similar events?

The implication here being that, rather than quanta of mass or energy being the limit of reductionism, the foundational reality in the universe (or one strata of it) might be order. The logos, if you like, or the Platonic ideal, or perhaps Kepler’s Sacred Geometry.

 

[Enai de a lukal]

And I see that as simply false. Languages are not objects.

Can you elaborate on this distinction? Obviously there are senses of the term in which languages can be objects- they can be subjects of true declarative sentences, for instance- so I'm wondering what is the relevant sense in which they are not?

A couple of comments.

1. Mathematics can, and does, look at 'inconsistent structures'. Paraconsistent logic allows some statements and their negations to both be true. However, unlike standard logic, a contradiction doesn't prove every statement (and thereby trivialize the structure).

2. I'm not so sure the structuralist and the formalist approach are that different. I look at axiom systems and what can be shown from them. The point is that every axiom system is a 'structure' and all definable structures are axiom systems. So I think the main difference is one of ontology. Do those 'structures' actually 'exists' as structures? Or are they actually just collections of assumptions and conclusions with no other ontology?

Again, the game of chess is a structure. To what extent and how does it 'exist'? I ultimately see the game of chess as a (rather simple) axiom system which doesn't allow for much model building.

Structuralism and formalism aren't all that different, they're definitely kindred spirits if nothing else. Both are an anti-realist/nominalist (or anti-realist adjacent, at any rate) position- there aren't robust mathematical/abstract objects in the usual sense, but something more minimal; either just the literal symbols themselves (formalism) or some form of fictitious concept (number, set, etc) that serves some purpose. But that's the question this brings to mind- you've expressed your view that mathematics are discovered, not invented. This is the standard realist/Platonist/etc position; there are mathematical/abstract objects, and there are truths concerning those objects which we discover rather than stipulate. But so then this is in tension with your expressed support for anti-realist philosophies of mathematics. Maybe a terminological confusion?

 

[muhammad_isa]

Abstractions exist, they just aren't real.

..you mean that they do not consist of matter .. "real" is not confined to a physical object .. is it?

 

[Polymath257]

Well, saying that just implies that revelations/scrolls that are claimed to have been from the One of God of Abraham, are all part of one gigantic conspiracy .. no?

No. There are many different traditions concerning how sacred texts are written. For the most post they are written by believers with the goal of motivating others to join their cause. That isn’t conspiracy, just good marketing.

You can answer in the context of the major revelations as a starting point.
i.e. the OT, NT & Qur'an

Well, in all cases we know of parts of the text that are simply wrong. That alone shows them not to be from a supreme deity. We also know that the texts were not written by those who they are traditionally attributed to.

Add to that the obvious motivation of priests to convince people to join and we see the result.

If not, I assume you have an agenda to confuse the issues raised, by bringing up every Tom, Dick & Harry.

Not at all. I just don’t see the texts as being strong evidence in any sense.

 

[muhammad_isa]

No. There are many different traditions concerning how sacred texts are written. For the most post they are written by believers with the goal of motivating others to join their cause..

..so these believers who wrote these texts were telling the truth, or were they lying, or something else?

That isn’t conspiracy, just good marketing..

Marketing? What do you think they were selling??

Well, in all cases we know of parts of the text that are simply wrong.

In some cases, yes .. not so much the underlying theme (about G-d), but in the details.

That alone shows them not to be from a supreme deity..

Not all the texts are even CLAIMED to be written or dictated by G-d.
Many of them are claimed to be inspired by G-d .. particularly the NT.

We also know that the texts were not written by those who they are traditionally attributed to.

We do, when in comes to the Bible .. but not the Qur'an .. we know no such thing.
This does not make the Bible wrong, or fraudulent .. it just makes it less reliable, imo.

Add to that the obvious motivation of priests to convince people to join and we see the result.

When it comes to creeds, then yes, I would agree.
..but not when it comes to the Psalms .. or the Gospels, or the Qur'an.

I just don’t see the texts as being strong evidence in any sense.

No. There could be many reasons for that .. and G-d knows better than you or I about
our own selves. Our intentions are often not even totally clear to us. We have complex minds.

 

[Heyo]

..you mean that they do not consist of matter .. "real" is not confined to a physical object .. is it?

They do not consist of matter or energy, or are fields. Real is what can be measured with a scientific instrument.

 

[muhammad_isa]

Real is what can be measured with a scientific instrument.

Right .. you are using the word "real" in that context.

 

[Polymath257]

..you mean that they do not consist of matter .. "real" is not confined to a physical object .. is it?

The term 'matter' usually refers to things made out of atoms, so isn't appropriate for subatomic particles, light, and many other things we consider to be 'physical'.

There are certainly several different definitions of the term 'real'. The only one that seems appropriate here relates to 'actually existing', which means it would be confined to 'physical objects'.

The issue I have is that the term 'physical' reduces to 'anything that interacts with something we already consider to be physical'. So, for example, light is physical, as are neutrinos, and electromagnetic fields.

Ultimately, I think the term 'non-physical' is incoherent: anything that actually exists supervenes on the physical: for there to be a change in *anything*, there needs to be a change in the physical. Ultimately, this is because any interaction would, eventually, be considered to be a physical one.

 

[Polymath257]

Right .. you are using the word "real" in that context.

And what do you mean by it?

 

[Polymath257]

..so these believers who wrote these texts were telling the truth, or were they lying, or something else?

I don't think it is that simplistic. People are story tellers. They also can be wrong about their own experiences in how they interpret them.

Some of the stories were probably told to make sense out of events around them. Some were told to support the power of priests and monarchs. Some were told to explain away failures or setbacks.

Once people started thinking of the stories as inspired, the road to a religious position is pretty straight forward. That doesn't require anyone to lie. But it may well be that they were simply wrong about what actually happened.

Also, some stories were acquired from cultures around them, or when the culture split off from others (many old testament stories have roots to Canaanite religion).

Marketing? What do you think they were selling??

The power of priests and monarchs. Sometimes the two were in competition, so you get propaganda and marketing from both sides.

In some cases, yes .. not so much the underlying theme (about G-d), but in the details.

Every culture makes up stories to explain the world around them. The Abrahamic religions seem to be no different on that score than any others.

Not all the texts are even CLAIMED to be written or dictated by G-d.
Many of them are claimed to be inspired by G-d .. particularly the NT.

Precisely. Some people *felt* inspired. Others found the stories to be inspiring. But having an inspiring story doens't mean it is factual or has a supernatural origin.

We do, when in comes to the Bible .. but not the Qur'an .. we know no such thing.
This does not make the Bible wrong, or fraudulent .. it just makes it less reliable, imo.

The Quran was written down long after Mohammed (IIRC, under the Ummayaad caliphs). Similarly for the hadiths.

When it comes to creeds, then yes, I would agree.
..but not when it comes to the Psalms .. or the Gospels, or the Qur'an.

Some of the Psalms are songs sung during religious services, so yes, those were to support the power of the priests. Others were telling people the basic rules of the society--how to act and what to consider to be important. The same is true of the Quran (power of Mohammed and his clan).

The gospels were written fairly long after the events they relate, mostly after the legend had grown, and not by those who they are usually attributed to.

No. There could be many reasons for that .. and G-d knows better than you or I about
our own selves. Our intentions are often not even totally clear to us. We have complex minds.

This assumes that there is a deity. It is NOT evidence for a deity. Nor is it evidence of a supernatural. it is a claim, and not much more.

Now, I think we should probably go to a different thread for this part of the discussion. It isn't directly related to whether math is discovered or invented. I am more than willing to debate in another thread, though.

 

[Polymath257]

By physical, for the sake of this argument let’s say possessing definite material properties. And by fundamental, let’s say at the limit of reductionism; or, looked at from a different angle, that from which all things emerge.

If subatomic particles have properties only in the manner of their relation to other particles and to the forces governing those relations; and if they don’t even have continuous temporal trajectories, but only the probability of materialising at any given set of co-ordinates from one moment to another, can they be said to exist as entities at all? Or are they rather, spontaneous events (as Schrodinger described them) unfolding in web of similar events?

Well, that partly depends on what you mean by an 'entity'.

I have maintained that we need to overhaul metaphysics because of the discoveries of quantum mechanics. The classical notions (substance, entities, causality, material, and many others) simply don't apply in the real world at that level.

The implication here being that, rather than quanta of mass or energy being the limit of reductionism, the foundational reality in the universe (or one strata of it) might be order. The logos, if you like, or the Platonic ideal, or perhaps Kepler’s Sacred Geometry.

No. The 'foundation' is particle fields that give probabilities for different things to happen. Order comes out of interactions, not the other way around.

Sorry, but Kepler was writing long before we understood many aspects of our solar systema nd the larger universe. he was instrumental for one crucial step, but his sacred geometry (platonic solids between the planetary orbits) is simply laughable now.

I consider 'Platonic ideals' to be one of the great mistakes of classical philosophy. I much prefer Aristotle and his skepticism.

 

[Polymath257]

Can you elaborate on this distinction? Obviously there are senses of the term in which languages can be objects- they can be subjects of true declarative sentences, for instance- so I'm wondering what is the relevant sense in which they are not?

Structuralism and formalism aren't all that different, they're definitely kindred spirits if nothing else. Both are an anti-realist/nominalist (or anti-realist adjacent, at any rate) position- there aren't robust mathematical/abstract objects in the usual sense, but something more minimal; either just the literal symbols themselves (formalism) or some form of fictitious concept (number, set, etc) that serves some purpose. But that's the question this brings to mind- you've expressed your view that mathematics are discovered, not invented. This is the standard realist/Platonist/etc position; there are mathematical/abstract objects, and there are truths concerning those objects which we discover rather than stipulate. But so then this is in tension with your expressed support for anti-realist philosophies of mathematics. Maybe a terminological confusion?

Once again, consider the game of chess. It was invented. The rules are ones that someone made up.

But, it is possible to have 'chess problems' and we can *discover* the solutions to those problems.

I think that the rules of mathematics are invented to help us model the world around us and the consequences of those rules are discovered after that invention.

I am certainly NOT a Platonist in any sense of the term.

 

[Polymath257]

Well, saying that just implies that revelations/scrolls that are claimed to have been from the One of God of Abraham, are all part of one gigantic conspiracy .. no?

You can answer in the context of the major revelations as a starting point.
i.e. the OT, NT & Qur'an

The major religions are simply the stories we still tell and people believe. We no longer believe the stories of Zeus and Athena. But they were believed at one time. That doesn't make them 'great conspiracies', just stories people told and believed that were wrong.

If not, I assume you have an agenda to confuse the issues raised, by bringing up every Tom, Dick & Harry.

I'm not sure what issues you see. Just like the stories from ancient Egypt, or Greece, or Norse mythology, or any number of other *religious* systems, the stories were told, some people started to believe them and they were passed down as *necessary* to believe.

I see the Abrahamic religions as just one branch of a very complex human endeavor to understand the world around us. This endeavor has had many false starts and dead ends. People believe for a variety of reasons, not at all limited to proper skepticism and investigation. Emotions get involved and, since the religions are seen as holding up the value system and, thereby, the society at large, any questioning is seen as dangerous and discouraged (often violently).

No 'conspiracy' is required: simply belief and being mistaken and superstitious.

 

[muhammad_isa]

Ultimately, I think the term 'non-physical' is incoherent: anything that actually exists supervenes on the physical: for there to be a change in *anything*, there needs to be a change in the physical..

It's irrelevant.
..and there does not necessarily need to be a change in "the physical".
..such as "I've changed my mind" doesn't mean that I have a new brain.

 

[Polymath257]

It's irrelevant.
..and there does not necessarily need to be a change in "the physical".
..such as "I've changed my mind" doesn't mean that I have a new brain.

It does mean some neutron fired that wouldn’t have otherwise, so there *is* a physical change. As to whether it is a “new brain”, the ship of Theseus comes to mind.

Once again, we should probably take this to a different thread.

 

[muhammad_isa]

Every culture makes up stories to explain the world around them. The Abrahamic religions seem to be no different on that score than any others..

That's patently untrue.
..unless of course, you consider yourself more educated, or more smart than "the average bear".

That is, you consider believers to be gullible and/or foolish.

The Quran was written down long after Mohammed (IIRC, under the Ummayaad caliphs). Similarly for the hadiths.

So what?
What exactly has changed in meaning, would you say?
There is a reason why it wasn't initially written down .. yes, you know that .. it was memorised.
Still is.

Some of the Psalms are songs sung during religious services, so yes, those were to support the power of the priests..

Too simplistic .. we don't need priests to believe in G-d.

Now, I think we should probably go to a different thread for this part of the discussion. It isn't directly related to whether math is discovered or invented. I am more than willing to debate in another thread, though.

As you like..